Turn off power to the air handler.
Turn off power to the condenser.
Remove the 2 BROWN wires from the delay on break relay.
Join them together with a wire nut.
This will send the COMMON from the transformer directly to the contactor.
Do the same with the 2 YELLOW wires on the delay on break relay.
This sends YELLOW(low voltage line) to the contactor.
Turn power on at air handler and condenser.
Set the thermostat to COOL.
Wait up to 5 minutes for the thermostat to reset and send power to the contactor.
Use your volt meter set for AC volts to check for 24 volts at the contactor BROWN and YELLOW wires.
If this works you can leave the delay relay out of the circuit because the digital stat will provide the time delay.

Thanks. Yes, the wire coming from the house/thermostat is brown (bottom left of my drawing). It breaks into a red and a black wire, where the red goes to the Delay On Break and the black goes to the "freon valve." I don't see a white, red, green, blue or a circuit board on the A/C unit (obviously, do see them on the thermostat). Will take a picture and and post it.

What I labeled "freon valve" may not be a valve at all. It is 2 yellow wires going into a junction on a copper tube where, I assume, freon flows. Will take a pic of that as well.

Would it just be worth swapping out the "delay on break" / short circuit timer or should I wait to test it by bypassing it first?

Turn off power to the air handler.Turn off power to the condenser.Remove the 2 BROWN wires from the delay on break relay.Join them together with a wire nut.This will send the COMMON from the transformer directly to the contactor.Do the same with the 2 YELLOW wires on the delay on break relay.This sends YELLOW(low voltage line) to the contactor.Turn power on at air handler and condenser.Set the thermostat to COOL.Wait up to 5 minutes for the thermostat to reset and send power to the contactor.Use your volt meter set for AC volts to check for 24 volts at the contactor BROWN and YELLOW wires.If this works you can leave the delay relay out of the circuit because the digital stat will provide the time delay.

Do this.The delay on break is not necessary with a digital thermostat.

For one that's a heat pump. That "valve" must be the reversing valve. You have low and high pressure switches. The HP switch has a reset button next to the suction line connection on the back of the unit. There's a red rubber button sticking out. Push in on it and see if it makes a click. Your LP switch should only be out if you lost most of your freon. Pretty easy to depress the pressure port strader valve to make sure you still have good pressure. If either switch is open no 24v signal reaches the contactor. I agree; bypass the delay timer.